On an Open Question Concerning Product-Type Difference Equations

TL;DR

This paper investigates the behavior of solutions to a specific system of difference equations when the parameter p is even, answering an open question about the existence of periodic solutions based on parameter relationships.

Contribution

It provides a detailed analysis of the solution behavior for even p, clarifying conditions under which solutions are periodic or not, extending previous work that focused on odd p.

Findings

Solution periodicity depends on coprimality of p and q.

Solutions may or may not be periodic based on p's parity.

The parity of p divided by gcd(p,q) influences solution behavior.

Abstract

In [Acta Math. Univ. Comenianae Vol. LXXX, 1 (2011), pp. 63--70], Yang, Chen and Shi examined the system of difference equations \[ x_n=\frac{a}{y_{n-p}},\qquad y_n=\frac{by_{n-p}}{x_{n-q}y_{n-q}},\qquad n=0,1,\ldots, \] where $q$ is a positive integer with $p < q$, $p \nmid q$, $p \geq 3$ is an odd number, both $a$ and $b$ are nonzero real constants, and the initial values $x_{-q+1},x_{-q+2},\ldots,$ $x_0,y_{-q+1},y_{-q+2},\ldots,y_0$ are nonzero real numbers. At the end of their note, they posted a question regarding the behaviour of solutions of the given system when $p$ is even. More precisely, they asked what the solutions of the system may look like if $p$ is even. In this note we answer this question raised by the authors. Particularly, we show that the system may or may not admit a periodic solution depending on the coprimality of the parameters $p$ and $q$ and on the parity of the integer $p/\gcd(p,q)$.